From each of two given numbers, half the smaller number is subtracted. After such subtraction, the larger number is 4 times as large as the smaller number. What is the ratio of the numbers ?

Options

A. `5 : 2`
B. `1 : 4`
C. `4 : 1`
D. `4 : 5`

Answer

To solve the problem step by step, let's denote the two numbers as \( x \) (the smaller number) and \( y \) (the larger number).

### Step 1: Understand the problem
We need to subtract half of the smaller number \( x \) from both numbers. After this subtraction, the larger number \( y \) is said to be 4 times the smaller number \( x \).

### Step 2: Set up the equation
After subtracting half of the smaller number \( x \) from both numbers, we have:
- The new smaller number: \( x - \frac{x}{2} = \frac{x}{2} \)
- The new larger number: \( y - \frac{x}{2} \)

According to the problem, the new larger number is 4 times the new smaller number:
\[ y - \frac{x}{2} = 4 \left(\frac{x}{2}\right) \]

### Step 3: Simplify the equation
Now we simplify the equation:
\[ y - \frac{x}{2} = 2x \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ y = 2x + \frac{x}{2} \]

### Step 5: Combine like terms
To combine the terms on the right side, we need a common denominator:
\[ y = 2x + \frac{x}{2} = \frac{4x}{2} + \frac{x}{2} = \frac{5x}{2} \]

### Step 6: Find the ratio
Now we have expressed \( y \) in terms of \( x \):
\[ y = \frac{5x}{2} \]
To find the ratio of \( x \) to \( y \):
\[ \text{Ratio} = \frac{x}{y} = \frac{x}{\frac{5x}{2}} = \frac{x \cdot 2}{5x} = \frac{2}{5} \]

Thus, the ratio of the two numbers \( x \) and \( y \) is:
\[ \text{Ratio of } x \text{ to } y = 2:5 \]

### Final Answer
The ratio of the numbers is \( 2:5 \).

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